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The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.

Question: Are there other categories of algebraic objects that have interesting natural topologies that carry algebraic data like the Zariski topology on a ring (spectrum)? If they exist, what are they and how are they used?

Note: I wasn't really sure what to tag this, so I chose some general tags that seem to express the idea.

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Wonderful questions! I've had similar vague questions along these lines floating around in my head for a couple years now. Is there a deeper explanation of this "really interesting coincidence"? It seems to suggest that the definition of "topological space", which somehow always seemed kind of a weird definition to me, has some kind of deeper significance, since it pops up everywhere... –  Kevin H. Lin Feb 5 '10 at 22:21
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@Kevin: I think some of the mystery disappears if you think in terms of the Kuratowski closure axioms. Any relation R between a set S and a set T - any relation at all - defines a topology on S by setting the closure operation to be cl(a) = {b | forall t such that aRt, also bRt}. You get the Zariski topology when T is a ring, S its set of prime ideals, and the relation is vanishing. –  Qiaochu Yuan Feb 5 '10 at 22:38
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Ah, sorry, that should read cl(A) = {b | forall t such that (aRt forall a in A), also bRt}. This is because a relation defines a Galois connection between the subsets of S and the subsets of T and the composition of the two functions making up the Galois connection is a closure operator. –  Qiaochu Yuan Feb 5 '10 at 22:41
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@Qiaochu: The Zariski topology is a bit deeper than that. Ideal addition and multiplication commute with the operations in nontrivial ways. –  Harry Gindi Feb 5 '10 at 22:42
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Harry and Qiaochu: Thanks for your responses to my question, though I don't feel like I fully understand either of your answers. I'd like to hear you guys elaborate on this though. I welcome you to email me on this matter. (Or perhaps I should make a new question on this.) BTW, I just noticed this interesting comment of Allen Knutson on Grothendieck, who apparently thought that the definition of topological space is "wrong": mathoverflow.net/questions/8204/… –  Kevin H. Lin Feb 6 '10 at 7:55
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7 Answers

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Yes, there are plenty of such things.

[In the following, "compact" implies "locally compact" implies "Hausdorff".]

1) To a Boolean algebra, one associates its Stone space, a compact totally disconnected space.

(Via the correspondence between Boolean algebras and Boolean rings, this is a special case of the Zariski topology -- but with a distinctive flavor -- that predates it.)

2) To a non-unital Boolean ring one associates its Stone space, a locally compact totally disconnected space.

3) To a commutative C*-algebra with unit, one associates its Gelfand spectrum, a compact space.

4) To a commutative C*-algebra without unit, one associates its Gelfand spectrum, a locally compact space.

6) To a commutative Banach ring [or a scheme over a non-Archimedean field, or...] one associates its Berkovich spectrum (the bounded multiplicative seminorms).

7) To a commutative ring R, one associates its real spectrum (prime ideals, plus orderings on the residue domain.)

8) To a field extension K/k, one associates its Zariski Riemann surface (equivalence classes of valuations on K which are trivial on k).

This is by no means a complete list...

Addendum: I hadn't addressed the second part of your question, i.e., explaining what these things are used for. Briefly, the analogy to the Zariski spectrum of a commutative ring is tight enough to give the correct impression of the usefulness of these other spectra/spaces: they are topological spaces associated (cofunctorially) to the algebraic (or algebraic-geometric, topological algebraic, etc.) objects in question. They carry enough information to be useful in the study of the algebraic objects themselves (sometimes, e.g. in the case of Stone and Gelfand spaces, they give complete information, i.e., an anti-equivalence of categories, but not always). In some further cases, one can get the anti-equivalence by adding further structure in a very familiar way: one can attach structure sheaves to these guys and thus get a class of "model spaces" for a certain species of locally ringed spaces -- e.g., Berkovich spectra glue together to give Berkovich analytic spaces.

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Heh, you actually put compact Hausdorff in your answer. You've been caught red-handed! –  Harry Gindi Feb 5 '10 at 22:01
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@HG: Oops. The oversight has been corrected. May Nicolas B. forgive me. –  Pete L. Clark Feb 5 '10 at 22:06
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as an addendum, and at the risk of mention "uncool" parts of functional analysis: in 3) and 4) you can replace $C^*$-algebra with "Banach algebra" -- although then the canonical map from an algebra A to C_0(Gelf(A)) might not be injective and is usually not isometric. Note that the analogy/contrast with usual comm. algebra is that we use the max ideal spectrum not the prime ideal spectrum. I'm not sure how this reconciles with 6) ... –  Yemon Choi Feb 5 '10 at 22:07
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Harry, have you come across Peter Johnstone's book Stone Spaces? It covers most of Pete's examples, and does almost everything in a categorical way. –  Tom Leinster Feb 5 '10 at 22:59
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@Pete, this is a wonderful answer. I'd like to nominate it for best of MO (I saw someone else nominate a post like this recently. Is that actually something we can do, or was he just using it as a rhetorical device?) –  Harry Gindi Feb 6 '10 at 0:46
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The $I$-adic topology on a commutative ring $A$ (with unity), where $I$ is an ideal of $A$. The closed sets are intersections of finite unions of sets of the form $a+I^n$ with $a\in A$ and $n\in\mathbb{N}$ (where $\mathbb{N}$ includes $0$). This topology has many trivial but very useful properties such as: The ring $A$ is separated (=Hausdorff) with respect to this topology if and only if $\displaystyle\bigcap_{n\in\mathbb{N}}I^n=0$. The most important example is the polynomial ring $A=B\left[X_1,X_2,...,X_n\right]$ with the ideal $I=\left(X_1,X_2,...,X_n\right)$. This one is separated, but not complete. Its completion is the ring of power series $B\left[\left[X_1,X_2,...,X_n\right]\right]$.

This is probably the most elementary example of a topology in algebra. I think Szamuely's book has more advanced ones.

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Krull topology on pro-finite completions of groups is perhaps of the same kind. –  Anweshi Feb 5 '10 at 22:44
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Interesting questions. Actually, this is indeed related to work on defining a natural topology on categories, which is part of noncommutative algebraic geometry.

A. Rosenberg defined the left spectrum for a noncommutative ring in 1981 (see The left spectrum, the Levitzki radical, and noncommutative schemes), and further generalized this spectrum to any abelian category (see reconstruction of schemes), and proved the so called Gabriel-Rosenberg reconstruction theorem which led to the correct definition of noncommutative scheme. I might have time to talk about this later. But for now, I shall just point out some papers, such as Spectra of noncommutative spaces.

In this paper, Rosenberg takes an abelian category as a "noncommutative space" and defines various spectra for different goals. (ONE remarkable destination is for representation theory of Lie algebras and quantum groups.)

One can not only define spectrum for abelian categories; this notion also makes sense in a non-abelian category and a triangulated category. In the paper Spectra related with localizations, Rosenberg defined the spectrum directly related to localization of categories. Roughly speaking, the spectrum of a category is a family of topologizing subcategories (which by definition, are closed under direct sum, sub- and quotient; in particular, thick or Serre subcategories) satifying some additional conditions.

There is also another paper, Underlying spaces of noncommutative schemes, trying to investigate the underlying space of a noncommutative scheme or other noncommutative "space" in noncommutative algebraic geometry. If we want to save flat descent in general, we might lose the base change property. In this work, Rosenberg deals with the "quasi-topology" (which means dropping the base change property) and defines the associative spectrum of a category. Moreover: for the goals of representation theory, he built a framework relating representation theory with the spectrum of abelian category (in particular, categories of modules). Actually, in this language, irreducible representations are in one-to-one correspondence with the closed points in the spectrum; generic points in the spectrum also produce representations (not necessarily irreducible).

The most important part in this work is that it provided a completely categorical (algebro-geometric) way to do induction in an abelian category instead of the derived category. (I will explain this later if I have time). This semester, Rosenberg gave us a lecture course, using this framework to compute all the irreducible representations for the Weyl algebra, the enveloping algebra, quantized enveloping algebras, algebras of differential operators, $SL\_2({\mathbb R})$ and other algebraic groups, or related associative algebras. It works very efficiently. For example, computing irreducible representations of $U(sl_3)$ is believed to be very complicated, but using this spectrum framework, it becomes much simpler.

The general framework for these is contained in the paper Spectra, associated points and representation theory. If you want to see some concrete examples using this machine, you should look at Rosenberg's old book Noncommutative Algebraic Geometry And Representations Of Quantized Algebras. There is another paper Spectra of `spaces' represented by abelian categories, providing the general theory for this machinery.

Furthermore, we can define the spectrum for an exact category; even more generally, for any Grothendieck site, and so for any category (because any category has a canonical Grothendieck pretopology). Rosenberg has recent work defining the spectrum for such categories -- Geometry of right exact `spaces' -- the main motivation for this work is to provide a background for higher universal algebraic K-theory for a right exact category (a category with a family of strict epimorphisms can be taken as a one-sided exact category). More important motivation is to study algebraic cycles for noncommutative schemes. (Warning: this paper is very abstract and hard to read. We will go through this paper in the lecture course this semester.)

All of these things will appear soon in his new book with Konstevich (but I am not sure of the exact time). If I have enough time to post, I will explain in more detail, how the theory of the spectrum for abelian categories comes into representation theory, and how this picture is related to the derived picture of Beilinson-Bernstein and Deligne. In fact, today we have just learned Beck's theorem for Karoubian triangulated categories and will do the DG-version of Beck's theorem later. And then he will introduce the spectrum for triangulated categories, and explain the noncommutative algebraic geometry facts behind the BBD machine and the connection with his abelian machine.

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Why did you do that with the text? +1, but oww, my eyes. –  Harry Gindi Feb 6 '10 at 1:52
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I've taken the liberty of editing the English in this answer/diary entry, and tweaking some of the BOLDFACE formatting. Hopefully it is now a bit easier to read; it has lots of interesting detail and the author is evidently very enthusiastic about this programme of research, so I felt it was a shame left as it was. –  Yemon Choi Feb 7 '10 at 9:59
    
Unfortunately the links format at MPI preprint server changed. For example previous mpim-bonn.mpg.de/preprints/send?bid=3617 does not work but instead mpim-bonn.mpg.de/preblob/3617 Therefore after the mpi website name one writes preblob/number instead of preprints/send?bid=number –  Zoran Skoda May 12 '11 at 17:23
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To any first order structure you can associate a Zariski-like topology, roughly by taking as closed sets the subsets definable by formulas without negation, see e.g. here and in the article linked there.

If the first order structure is an algebraically closed field where you interpret the language of rings you get back the Zariski topology.

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Interesting finite groups tend to have interesting inherent geometries (just as orbit-stabilizer turns external actions into internal actions, similar ideas turn many external geometries into coset geometries). The geometry induced by conjugation on Sylow p-subgroups is important for all finite groups, and turns out to describe the (p-completion of the) classifying space of the group.

Geometry has always been an important part of group theory. Zassenhaus groups and sharply triply transitive groups typically have an underlying affine or projective plane they are acting on. Early investigations of these special permutation groups in the 1930s led to some of the systematic development of finite geometry over things other than fields. You can recover the algebraic structure of something like a ring just from the permutation action of the group (often on a regular subgroup). M. Hall Jr.'s textbook on the Theory of Groups has a nice exposition of these ideas.

Of course finite groups of Lie type acting on their Borel subgroups also define important geometries, roughly called "buildings", and there are a great many references for those. This became a very popular way to understand the non-sporadic groups. These groups of Lie type have other nice actions, often on interesting finite geometries called generalized polygons.

Equivariant homotopy people noticed that some of these geometries are nearly enough to define a classifying space of the group, along with a nice decomposition of its cohomology ring. D. Benson and S.D. Smith's book on Classifying Spaces of Sporadic Simple Groups (MR2378355) describes these techniques with a reasonably algebraic feel. Modulo a few details, these are the fusion systems Scott Carnahan mention in a previous thread, MO5659. These geometries were investigated in order to provide a more natural analogue of buildings for sporadic groups.

Actually, I suppose you might feel that classifying spaces themselves are naturally associated to finite groups.

Edit: I thought it might be helpful to point out the similarities to the Zariski topology: The Zariski topology basically encodes how prime ideals intersect. The fusion of a finite group encodes how Sylow subgroups intersect. Strong fusion not only keeps track of the intersections, but also of the (G-inner) maps between those intersections, so that the fusion becomes a category. Since fusion controls cohomology, it seemed natural to look at how fusion describes the classifying space of a group. Amazingly, it does a great job of describing the p-completion of the classifying space and facilitates fairly direct calculations. In other words, the data encoded by the "prime subgroups" (Sylow p-subgroups) also encodes a natural topological space associated to the group, its (p-completed) classifying space.

Several areas of combinatorics, like certain parts of graph theory and finite geometry, also seem to be based on the simple fact that interesting groups have interesting geometries. A recent classification of Steiner triple systems followed from detailed classifications of finite simple groups and multiply transitive permutation groups, and several families of graphs are interesting because of their automorphism groups.

I hope it is clear too that separating a group from its actions is not sensible. The actions of a group are encoded by the conjugacy classes of its subgroups, and it is entirely internal. Most geometries associated to groups are also internal. This is basically why the classification of finite simple groups can succeed: the natural action of a group is already contained inside it in an easy to describe way, so that once the local structure of a group is sufficiently similar to a known group, the group itself is isomorphic to a known group.

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carry algebraic data like the Zariski topology on a ring (spectrum)? If they exist, what are they and how are they used?

In model theory they define and study the 'algebraic data like the Zariski topology' irrespectively of where these data come from. These data are called Zariski geometries, and e.g. admit some intersection theory, can be used to prove Chow's lemma, admit some classification results in dim 1 etc. You may want to have a look on the recent book of Zariski geometries and references therein (or the actual book Zariski Geometries : Geometry from the Logician's Point of View, by Boris Zilber).

Also, the book has some examples which sometimes need some work.

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Given a group theoretic class $\mathfrak{X}$ (e.g., finite groups, soluble groups, etc.), to each group $G$ one can associate the pro-$\mathfrak{X}$ topology on $G$ by taking as a basis of neighbourhoods of the identity the collection of normal subgroups $N$ of $G$ for which the quotient group $G/N$ belongs to $\mathfrak{X}$. A group is residually an $\mathfrak{X}$-group precisely when this topology is Hausdorff. (To get an actual topology, $\mathfrak{X}$ has to be hereditary and closed under (finite) direct products.)

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